{
  "id": "2104.04783",
  "version": "v1",
  "published": "2021-04-10T14:43:56.000Z",
  "updated": "2021-04-10T14:43:56.000Z",
  "title": "A Class Of Curvature Flows Expanded By Support Function And Curvature Function In The Euclidean Space And Hyperbolic Space",
  "authors": [
    "Shanwei Ding",
    "Guanghan Li"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\\mathbb{R}^{n+1}$ with speed $u^\\alpha f^{-\\beta}$, where $u$ is the support function of the hypersurface, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For $\\alpha \\le 0<\\beta\\le 1-\\alpha$, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt \\cite{GC3} and Urbas \\cite{UJ2} can be recovered by putting $\\alpha=0$ and $\\beta=1$ in our first result. If the initial hypersurface is convex, this is our previous work \\cite{DL}. If $\\alpha \\le 0<\\beta< 1-\\alpha$ and the ambient space is hyperbolic space $\\mathbb{H}^{n+1}$, we prove that the flow $\\frac{\\partial X}{\\partial t}=(u^\\alpha f^{-\\beta}-\\eta u)\\nu$ has a longtime existence and smooth convergence to a coordinate slice. The flow in $\\mathbb{H}^{n+1}$ is equivalent (up to an isomorphism) to a re-parametrization of the original flow in $\\mathbb{R}^{n+1}$ case. Finally, we find a family of monotone quantities along the flows in $\\mathbb{R}^{n+1}$. As applications, we give a new proof of a family of inequalities involving the weighted integral of $k$th elementary symmetric function for $k$-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in \\cite{GL2}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-10T14:43:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic space",
      "support function",
      "euclidean space",
      "curvature flows",
      "curvature function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}